1. Field
Embodiments described herein relate generally to a blood flow perfusion analyzing apparatus, a blood flow perfusion analyzing method, a fluid analyzing apparatus and a fluid analyzing method.
2. Description of the Related Art
Conventionally, an examination in which contrast medium is injected from a vein, time series image data is acquired and the image data is analyzed to obtain information on a blood flow perfusion in tissue is performed with an X-ray CT (Computed Tomography) apparatus or an MRI (Magnetic Resonance Imaging) apparatus. This examination is called perfusion examination using the fact that a degree of concentration of contrast medium in an imaging section can be acquired as concentration changes in an image.
For example, a TCC, (Time Concentration Curve) corresponding to arterial blood flowing into tissue is set to an input function ai, deconvolution of the input function with a measured TCC Ci of the tissue is performed, an impulse response function (residue function) Ri inherent in the tissue is calculated and parameters including a blood flow (CBF: Cerebral Blood Flow in case of a brain) which is an index representing a blood flow perfusion quantitatively a MTT (Mean Transit Time) and a blood volume (CBV: Cerebral Blood Volume in case of a brain) based on the calculated impulse response function in order to reduce a fluctuation in a pulmonary circulation and a contrast medium infection. This analysis method is called a deconvolution method.
Note that a TCC is a curve representing a time change of a quantity in proportion as a density of contrast medium. A time change curve of quantity in proportion as a density of contrast medium obtained based on contrast image data acquired by an X-ray CT apparatus is also referred to a TDC (Time Density Curve). However, it is indicated as a TCC here.
For example, non-patent document 1 applies standard SVD (Singular Value Decomposition) method or block circulant SVD method to the deconvolution method.
Here, the standard. SVD method will be described. First, a change in pixel value at each pixel j (position ri) in acquired image data is measured to calculate a TCC Ci(rj) corresponding to a tissue. Then, a TCC Ai corresponding to an artery is calculated based on the calculated tissue TCC Ci as shown in following equation (1). In equation (1), i represents a phase number and ΩA represents a set of pixels corresponding to the artery.
                              A          i                =                              1            n                    ⁢                                    ∑                              j                ⋐                                  Ω                  A                                                                                                  ⁢                                                  ⁢                                          C                i                            ⁡                              (                                  r                  j                                )                                                                        (        1        )            
Assuming that a tissue TCC Ci is a deconvolution of an artery TCC Ai and an impulse response function X, the tissue TCC Ci can be expressed by a system equation as shown in following, equation (2). In equation (2), TR represents a time interval (TR: repetition time) imaging and Rj represents an impulse response function.
                              C          i                =                                            ∑                              j                =                0                            i                        ⁢                                                  ⁢                                          A                                  i                  -                  j                                            ⁢                              T                R                            ⁢                              R                j                                              =                                    ∑                              j                =                0                            i                        ⁢                                                  ⁢                                          A                                  i                  -                  j                                            ⁢                              X                j                                                                        (        2        )            
For example, assuming that data of L phases, which is derived by making the artery TCC Ai and the tissue TCC Ci discrete in the time axis direction, i.e. dividing a phase range of ibs≦i≦L−1 into L equal intervals, is used for the analysis, the above-mentioned equation (2) becomes C=AX in the matrix form and can be expressed as shown in following equation (3). In equation (3). C is a L×1 column vector, A is a L×1 coefficient matrix and X is a L×1 column vector.
                              (                                                                      C                                      i                    bs                                                                                                                        C                                                            i                      bs                                        +                    1                                                                                                      ⋮                                                                                      C                                                            i                      bs                                        +                    L                    -                    1                                                                                )                =                              (                                                                                a                                          0                      ,                      0                                                                                                            a                                          0                      ,                      1                                                                                        …                                                                      a                                          0                      ,                                              L                        -                        1                                                                                                                                                              a                                          1                      ,                      0                                                                                                            a                                          1                      ,                      1                                                                                        …                                                                      a                                          1                      ,                                              L                        -                        1                                                                                                                                          ⋮                                                  ⋮                                                                                                                                          ⋮                                                                                                  a                                          L                      -                      1.0                                                                                                            a                                          L                      -                      1.1                                                                                        …                                                                      a                                                                  L                        -                        1                                            ,                                              L                        -                        1                                                                                                                  )                    ⁢                      (                                                                                X                    0                                                                                                                    X                    1                                                                                                ⋮                                                                                                  X                                          L                      -                      1                                                                                            )                                              (        3        )            
In above-mentioned equation (3), ibs represents a phase at the head of a phase range to be analyzed and uses a phase at or before which the contrast concentrations in the artery and the tissue begin to rise. Each element value ai,j of the coefficient matrix is expressed by following equation (4). In equation (4) i=0, 1, . . . , L−1 and j=0, 1, . . . , L−1.
                              a                      i            ,            j                          =                  {                                                    0                                                                                                        for                      ⁢                                                                                          ⁢                      i                                        -                    j                                    <                  0                                                                                                      A                                      i                    -                    j                    +                                          i                      bs                                                                                                                                        for                    ⁢                                                                                  ⁢                    0                                    ≤                                      i                    -                    j                                    ≤                                      L                    -                    1                                                                                                          (        4        )            
Then, an approximate solution of above-mentioned equation (3) is calculated by a singular value decomposition. That is, the matrix A can be decomposed into 3 matrices as shown in following equation (5).A=UDVT  (5)
In above-mentioned equation (5), U is an orthonormal matrix having m rows and k columns, D is a diagonal matrix having k rows and k columns and V is an orthonormal matrix having n rows and k columns. When U, D and V are applied to above mentioned equation (3), m=n=k=L. A diagonal matrix is a matrix of which all elements other than diagonal elements are zero and an orthonormal matrix is a diagonal matrix of which diagonal elements are 1.
Assuming that U=(u1, u2, . . . , vk), V=(v1, v2, . . . , vk) and D=diag(w0, w1, . . . , wk), wk is sorted in descending order and subsequently uk and vk are also respectively sorted in the order corresponding to the order of the sorted wk. Then, the minimum normed estimated value which is the minimum square solution of X can be calculated with the result of the singular value decomposition of the matrix A by following equation (6).{tilde over (X)}=VWUTC  (6)
In above mentioned equation (6), W is a diagonal matrix consisting of the reciprocals of the diagonal elements (the singular values) of D and W=diag(1/w0, 1/w1, . . . , 1/wG-1, 0, 0, . . . ). When a singular value is below the maximum singular value times Psvd, not the reciprocal of the singular value but zero is set as a diagonal element. Therefore, W has G diagonal elements which are not zero. Psvd is a parameter which influences strength of the regularization. When Psvd is larger, a stronger regularization is performed.
Then, calculating the nonmed estimated value makes it possible to calculate the impulse response function Rj as shown in following equation (7).Rj={tilde over (X)}j/TR  (7)
A tissue blood flow volume, a tissue blood volume and a mean transit time can be calculated based on the impulse response function Rj calculated by above mentioned equation (7).
By the way, blood flows to capillaries in tissues. A time at which the contrast medium reaches a tissue depends on a place in the tissue, and therefore, the time axis of the artery TCC calculated with regard to a set arterial area is not same to that of the tissue TCC at each pixel strictly. Therefore, a contrast concentration in the tissue rises before that in the artery rises (the tissue TCC rises precedently before the artery TCC rises) in the extreme case.
In the above mentioned standard SVD method it is assumed that such a delay in the time direction does not occur. However, a deconvolution result changes depending on a delay time which is a difference in the time direction always existing between the artery TCC and tissue TCC. Consequently, the calculated tissue blood flow undergoes influence.
For that reason, the block circulant SVD method uses a system matrix which is a block circular matrix as shown in following equation (8) in order to reduce the dependence on the delay in the time direction.
                              (                                                                      C                                      i                    bs                                                                                                                        C                                                            i                      bs                                        +                    1                                                                                                      ⋮                                                                                      C                                                            i                      bs                                        +                                          2                      ⁢                      L                                        -                    1                                                                                )                =                                  ⁢                              (                                                                                b                                          0                      ,                      0                                                                                                            b                                          0                      ,                      1                                                                                        …                                                                      b                                          0                      ,                                                                        i                          bs                                                +                                                  2                          ⁢                          L                                                -                        1                                                                                                                                                              b                                          1                      ,                      0                                                                                                            b                                          1                      ,                      1                                                                                        …                                                                      b                                          1                      ,                                                                        i                          bs                                                +                                                  2                          ⁢                          L                                                -                        1                                                                                                                                          ⋮                                                  ⋮                                                                                                                                          ⋮                                                                                                  b                                                                                            i                          bs                                                +                                                  2                          ⁢                          L                                                -                        1                                            ,                      0                                                                                                            b                                                                                            i                          bs                                                +                                                  2                          ⁢                          L                                                -                        1                                            ,                      1                                                                                        …                                                                      b                                                                                            i                          bs                                                +                                                  2                          ⁢                          L                                                -                        1                                            ,                                                                        i                          bs                                                +                                                  2                          ⁢                          L                                                -                        1                                                                                                                  )                    ⁢                                                 (                                                                                          X                      0                                                                                                                                  X                      1                                                                                                            ⋮                                                                                                              X                                                                        2                          ⁢                          L                                                -                        1                                                                                                        )                                                          (        8        )                                                          ⁢                                            When              ⁢                                                          ⁢              i                        ≧                          j              ⁢                                                          ⁢                              b                                  i                  ,                  j                                                              =                      {                                                                                                                                                        A                                                      i                            -                            j                            +                                                          i                              bs                                                                                                                                                                                                                                      for                              ⁢                                                                                                                          ⁢                              i                                                        -                            j                                                    <                          L                                                                                                                                    0                                                                                              otherwise                          ⁢                                                                                                                                                                                      ⁢                                                                          ⁢                                                                          ⁢                  When                  ⁢                                                                          ⁢                  i                                <                                  j                  ⁢                                                                          ⁢                                      b                                          i                      ,                      j                                                                                  =                              {                                                                                                    A                                                  i                          -                          j                          +                                                      i                            bs                                                    +                          L                                                                                                                                                                                          for                            ⁢                                                                                                                  ⁢                            i                                                    -                          j                          +                          L                                                ≥                        0                                                                                                                        0                                                                                      otherwise                        ⁢                                                                                                                                                                                                                                          
The basic processing of the block circulant SVD method is similarly to that in the standard SVD method. That is, the matrix form of the block circulant SVD method is C=AX. However, A of the block circulant SVD method is a 2L×2L coefficient matrix. Accordingly, zeros are added rearward the respective element values constituting the tissue TCC in the column vector C to expand the coefficient matrix A, the column vector X and the column vector C to have double numerical elements of that in measured data. Especially, the coefficient matrix A is expanded to have double numerical rows and columns and above and rearranged so that elements in the column vector become periodic.
By composing a system matrix as shown in above mentioned equation (8), the impulse response function shifts in the time direction while keeping its shape a similar figure when a difference in the time direction exists. Therefore, shifting the impulse response function in the time direction makes it possible to calculate a tissue blood flow, a tissue blood volume and a mean transit time having reduced dependence on a delay in the time direction.
However, the size of the coefficient matrix A constituting the artery TCC becomes 2×2 times and the size of the column vector constituting the tissue TCC becomes double in the system matrix. Therefore, the time of analysis processing for the block circulant SVD method increases up to over 8 times and the block circulant SVD method is not practical.
For that reason, the non-patent document 2 puts the system matrix of the standard SVD method into practice without increasing the time of analysis processing. Specifically, the proposed method in the non-patent document 2 uses the system matrix shown by above mentioned equation (4) and excludes elements, which are within the head phase to an offset timing, in the tissue TCC from the known vector and calculation to compensate delay in this case, elements at some initial phases from the vector C to the offset timing are excluded from calculation by above mentioned equation (3) and instead of elements of which number is the same as that of the excluded elements are added to backward of the elements of the tissue TCC.
In this way, technique of the non-patent document 2 makes the dependence on the delay in the time direction small without extending the system matrix.
Because the technique of the non-patent document 1 does not consider the delay in the time direction with regard to the standard SVD method, the result of the deconvolution lacks sufficient accuracy. On the other hand, the block circulant SVD method requires much time for analysis processing though the delay in the time direction is considered. Therefore, it is a problem that both the standard SVD method and the block circulant SVD method are not practical.
Especially, X-ray CT apparatuses and MRI apparatuses are generally used for examinations of many patients. Further, analysis processing must be performed immediately after imaging and an operator must observe a result of the processing to determine whether the imaging has been performed appropriately or not. Therefore, if the time for analysis processing increases, the number of examinations which can be performed decreases, a waiting time of a patient increases and the number of patients processed by a certain hospital decreases. That is, increasing the time for analysis processing brings disadvantages to both patients and hospitals.
To such problem, the technique of the non-patent document 2 performs analysis processing without increasing processing time. However, a new problem occurs in that the processing result depends on an offset timing since elements, which are within the head phase to the offset timing, of the tissue TCC are excluded from calculation. Furthermore, it is difficult to identify the offset timing accurately. As a result, it is a problem that information on blood flow perfusion cannot be obtained stably.    [Non-patent document 1] Ona Wu et al, “Tracer Arrival Timing-insensitive. Technique for Estimating Flow in MR Perfusion-Weighted imaging Wine Singular Value Decomposition With a Block-Circulant Deconvolution Matrix”, Magnetic. Resonance in Medicine 50: 164-174 (2003)    [Non-patent document 2] M. R. Smith, H. Lu, S. Trochet, and R. Frayne “Removing the Effect of SVD Algorithmic Artifacts Present in Quantitative MR Perfusion Studies”, Magnetic Resonance in Medicine 51: 631-634 (2004)